Math: Completing the Square Worksheet

Question 1

Rewrite x^2 + 6x + 5 in the form (x + a)^2 + b.

Accepted Answer

The expression rewrites as (x + 3)^2 - 4 because x^2 + 6x + 5 = x^2 + 6x + 9 - 9 + 5.

Question 2

Rewrite x^2 - 10x + 7 in the form (x + a)^2 + b.

Accepted Answer

The expression rewrites as (x - 5)^2 - 18 because x^2 - 10x + 7 = x^2 - 10x + 25 - 25 + 7.

Question 3

Rewrite x^2 + 4x - 1 in the form (x + a)^2 + b.

Accepted Answer

The expression rewrites as (x + 2)^2 - 5 because x^2 + 4x - 1 = x^2 + 4x + 4 - 4 - 1.

Question 4

Rewrite x^2 - 8x + 3 in the form (x + a)^2 + b.

Accepted Answer

The expression rewrites as (x - 4)^2 - 13 because x^2 - 8x + 3 = x^2 - 8x + 16 - 16 + 3.

Question 5

Solve by completing the square: x^2 + 6x + 1 = 0.

Accepted Answer

The solutions are x = -3 + 2sqrt(2) and x = -3 - 2sqrt(2). Completing the square gives (x + 3)^2 = 8.

Question 6

Solve by completing the square: x^2 - 4x - 12 = 0.

Accepted Answer

The solutions are x = 6 and x = -2. Completing the square gives (x - 2)^2 = 16.

Question 7

Solve by completing the square: x^2 + 2x - 7 = 0.

Accepted Answer

The solutions are x = -1 + 2sqrt(2) and x = -1 - 2sqrt(2). Completing the square gives (x + 1)^2 = 8.

Question 8

Solve by completing the square: x^2 - 12x + 20 = 0.

Accepted Answer

The solutions are x = 10 and x = 2. Completing the square gives (x - 6)^2 = 16.

Question 9

Solve by completing the square: x^2 + 8x + 3 = 0.

Accepted Answer

The solutions are x = -4 + sqrt(13) and x = -4 - sqrt(13). Completing the square gives (x + 4)^2 = 13.

Question 10

Rewrite 2x^2 + 12x + 7 in vertex form a(x - h)^2 + k.

Accepted Answer

The expression rewrites as 2(x + 3)^2 - 11 because 2x^2 + 12x + 7 = 2(x^2 + 6x) + 7 = 2(x^2 + 6x + 9) + 7 - 18.

Question 11

Rewrite 3x^2 - 18x + 10 in vertex form a(x - h)^2 + k.

Accepted Answer

The expression rewrites as 3(x - 3)^2 - 17 because 3x^2 - 18x + 10 = 3(x^2 - 6x) + 10 = 3(x^2 - 6x + 9) + 10 - 27.

Question 12

Rewrite -x^2 + 4x + 1 in vertex form a(x - h)^2 + k.

Accepted Answer

The expression rewrites as -(x - 2)^2 + 5 because -x^2 + 4x + 1 = -(x^2 - 4x) + 1 = -(x^2 - 4x + 4) + 1 + 4.

Question 13

Solve by completing the square: 2x^2 + 8x - 6 = 0.

Accepted Answer

The solutions are x = 1 and x = -5. Dividing by 2 gives x^2 + 4x - 3 = 0, and completing the square gives (x + 2)^2 = 7.

Question 14

Solve by completing the square: 3x^2 - 6x - 9 = 0.

Accepted Answer

The solutions are x = 3 and x = -1. Dividing by 3 gives x^2 - 2x - 3 = 0, and completing the square gives (x - 1)^2 = 4.

Question 15

A ball's height is modeled by h(t) = t^2 - 6t + 13. Rewrite the function in vertex form and state the minimum height.

Accepted Answer

The function in vertex form is h(t) = (t - 3)^2 + 4. The minimum height is 4, which occurs at t = 3.

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